Nonparametric test One sample tests Two sample tests Testing for three or more samples

Transcription

1 Nonparametric test One sample tests Two sample tests Testing for three or more samples 1

2 Background So far we have stressed that in order to carry out hypothesis tests we need to make certain assumptions about the types of distributions from which we were sampling. For example. to do t tests we needed to assume that the populations involved were approximately normal. In the two sample t-test we needed to make the more specific assumption that the variances are equal. An important part of statistics deals with tests for which we do not need to make such specific assumptions. These tests are called nonparametric or distribution-free tests. These tests would ordinarily be used if a parametric test were not appropriate. This might happen. for instance. if you were working with a non normal distribution. or a distribution whose shape was not yet evident. It might also happen that you are working with some special type of data for which there was no appropriate parametric test HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach2 2

3 Ranking the data Nonparametric tests can't use the estimations of population parameters. They use ranks instead. Instead of the original sample data we have to use its rank. to show the ranking procedure suppose we have the following sample of measurements: Case 4 has the smallest value (68). it is assigned a rank of 1. Case 3 has the next smallest value. it is assigned a rank of 2. Cases 5 and 6 are equal. they are assigned a rank of 3.5. the average rank of 3 and 4. We say that case 5 and 6 are tied. The next table shows the result of ranking. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 3

4 Case Data Rank Tabulate the data n i = 1 r i = n( n 2 + 1) = 6*7 2 = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 4

5 One sample tests Sign test Wilcoxon sign test Two samples tests Type of tests (Mann-Whitney test) (Wilcoxon Rank-Sum test) More than two samples Kruskall-Wallis test Jonckheere-Terpstra test HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 5

6 Wilcoxon sign test Data are in pairs E.g.: before-after treatment We have n subjects and X (x 1.x 2...x n ). Y (y 1.y 2...y n ) denotes the variable before and after treatment. respectively. Ignore where x j =y j. x j =τ+ε i y j =τ- ν +ε i d j =x j -y j =ν + ε i -ε i E(d i )= ν ; and E(ε i )=E(ε i )=0 H 0 : ν=0 H a := ν>0; H a = ν<0 or H a ν 0 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 6

7 Wilcoxon Sign Test Calculate absolute values of z i. Sort them. Calculate δ i. The test statistics T + T δ i + 1, if = { 0, if = n' i = 1 d d i i > 0 < 0 δ ir i HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 7

8 Decision rule Use standard normal distribution table E( T ) = n( n + 1) 2 + n( n + 1)(2 + ; D ( T ) = n 1) z = T T + * = + D( T E( T ) + ) HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 8

9 Decision If the calculaterd z score is greater than 1.96, then Nullhypothesis is rejected, and the alternative hypothesis is accepted, namely the diffence is significant If the calculaterd z score is less than 1.96, then Nullhypothesis is accepted, namely the diffence is NOT significant. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 9

10 Standard normal probabilities z Φ(x): proportion of area to the left of Z HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 10 10

11 Example There is a treatment using a new drug at 9 patients. Data are summarised in the next table. X is the baseline hormone level Y is the after treatment hormone level Is there any changes at hormone levels after treatment? HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 11

12 The data i x i y i d i d i R i δ i δ i R i HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 12

13 H 0 : ν=0 H a ν 0 Test statistics T + n' = i = 1 δ ir i = 5 T α /2. n=9 =39 The intervall: T + 6 or T + 39 So we reject H 0 9* T Tα / 2, n= 9orT

14 STATA results sign obs sum ranks expected positive negative zero all unadjusted variance adjustment for ties 0.00 adjustment for zeros adjusted variance H o : xi = yi z = Prob > z = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 14

15 t-test: Paired Two Sample for Means before after Mean Variance Observations 9 9 Pearson Correlation df 8 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 15 15

16 Mann-Whitney Test (Non-parametric independent two-group comparisons) Definition: A non-parametric test (distribution-free) used to compare two independent groups of sampled data. Assumptions: Unlike the parametric t-test. this non-parametric makes no assumptions about the distribution of the data (e.g.. normality). Characteristics: This test is an alternative to the independent group t-test. when the assumption of normality or equality of variance is not met. This. like many non-parametric tests. uses the ranks of the data rather than their raw values to calculate the statistic. Since this test does not make a distribution assumption. it is not as powerful as the t-test. Test: The hypotheses for the comparison of two independent groups are: Ho: The two samples come from identical populations Ha: The two samples come from different populations HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 16 16

17 Mann-Whitney (M-W) procedure To compute the test. the observations from both samples are first combined and ranked from smallest to largest value. The statistic for testing the null hypothesis that the two distributions are equal is the sum of the ranks for each of the two groups. If the groups have the same distribution. their sample distributions of ranks should be similar. If one of the groups has more than its share of small or large ranks. there is reason to suspect that the two underlying distributions are different. If the total sample size is less than 30. tables can be used where an interval for R min -R max is given. If one of our test statistic is in the interval. we do not reject the null hypothesis. For large sample size a normal approximation is possible to get the p-value HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 17 17

18 M-W test Notice that the hypothesis makes no assumptions about the distribution of the populations. These hypotheses are also sometimes written as testing the equality of the central tendency of the populations. The test statistic for the Mann-Whitney test is U. This value is compared to a table of critical values for U based on the sample size of each group. If U exceeds the critical value for U at some significance level (usually 0.05) it means that there is evidence to reject the null hypothesis in favor of the alternative hypothesis. Note: Actually. there are two versions of the U statistic calculated. where U' = n 1 n 2 - U where n1 and n2 are the sample sizes of the two groups. The largest of U or U' is compared to the critical value for the purpose of the test. Note: For sample sizes greater than 8. a z-value can be used to approximate the significance level for the test. In this case. the calculated z is compared to the standard normal significance levels. Note: The U test is usually perform as a two-tailed test. however some text will have tabled one-tailed significance levels for this purpose. If the sample size if large. the z-test can be used for a one-sided test. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 18 18

19 Example (M-W) Professor Testum wondered if students tended to make better scores on his test depending if the test were taken in the morning or afternoon. From a group of 19 similarly talented students. he randomly selected some to take a test in the morning and some to take it in the afternoon. The scores by groups were: HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 19 19

20 The Data Morning Afternoon HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 20 20

21 Calculate ranks Morning Afternoon Morning Ranks Afternoon Ranks HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 21 21

22 Sum of ranks Σ Morning ranks = 119 Σ Afternoon ranks = 71 M-W critical value is є[75-125] So we accept null hypothesis. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 22 22

23 STATA Results of Mann-Whitney test Two-sample Mann-Whitney rank-sum test group obs rank sum expected combined unadjusted variance adjustment for ties adjusted variance Ho: data(group==1) = data(group==2) z = Prob > z = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 23 23

24 t-test: Two-Sample Assuming Equal Variances Morning Afternoon Mean 91,43 88,8 Variance 20, ,85 Observations 10 9 Pooled Variance 13, Hypothesized Mean Difference 0 df 17 t Stat 1, P(T<=t) one-tail 0, t Critical one-tail 1, P(T<=t) two-tail 0, HUSRB/0901/221/088 t Critical two-tail Teaching Mathematics and Statistics in Sciences: 2, Modeling and Computer-aided Approach 24 24

25 Wilcoxon Rank-Sum Test (Non-parametric independent two-group comparisons) Definition: A non-parametric test (distribution-free) used to compare two independent groups of sampled data. Test: The hypotheses for the comparison of two independent groups are: H 0 : The two samples come from identical populations H a : The two samples come from different populations HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 25 25

26 Wilcoxon Rank Sum test We have M=m+n observations in two groups: X (x 1.x 2...x m ). Y (y 1.y 2...y n ) denotes the variables. We suppose: x j =ε i i=1,2,..m y j =Δ+ε m+j, j=1,2,..., n x j,y j are the observed frequencies. H 0 : Δ =0 H a := Δ >0 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 26 26

27 Wilcoxon Rank-Sum Test Sort in ascending order the total M observations (Merge the two groups). If R j denotes the ranks of y j then calculate the sum of R j s. W = R j Test statistics (z) is approximately N(0,1) distributed for large M: n j = 1 z = W = W E( W D( W ) ) = W ( mn( n + n( m + m + n + 1) / 1) /12) 2 1/ 2 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 27 27

28 Example We have the following measurements of serum triglyceride level in two groups: Control (X; m=6) : Treated (Y; n=3): Conbine them and assign the ranks: HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 28 28

29 Example Conbine them and assign the ranks: X: Y: R: W= 1+2+5=8 Critical interval for W is [7-23] at α=0.05. Thus, we accept H 0. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 29 29

30 STATA Results of Wilcoxon ranksum test Two-sample Wilcoxon rank-sum (Mann-Whitney) test group obs rank sum expected control treated combined unadjusted variance adjustment for ties adjusted variance Ho: data(group_==0) = data(group==1) z = Prob > z = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 30 30

31 EXAMPLE After a randomised trial comparing aspririn with placebo for hadache, 8 patients on aspirin and 10 on placebo rated their improvement on a 10 cm kine. A measure of 0 indicating no improvement and one of 10 indicating very much better. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 31 31

32 Data Group Improvement Aspirin 7.5 Aspirin 8.3 Aspirin 9.1 Aspirin 6.2 Aspirin 5.4 Aspirin 8.3 Aspirin 6.5 Aspirin 8.4 Placebo 3.1 Placebo 5.6 Placebo 4.5 Placebo 6.2 Placebo 5.1 Placebo 5.3 Placebo 5.5 Placebo 4.1 Placebo 4.3 Placebo 4.2 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 32 32

33 Stata results Two-sample Wilcoxon rank-sum (Mann-Whitney) test mw_group obs rank sum expected Aspirin Placebo combined unadjusted variance adjustment for ties adjusted variance Ho: improvem(mw_group==aspirin) = improvem(mw_group==placebo) z = Prob > z = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 33 33

34 Kruskall-Wallis test We have more than two groups. (Non-parametric independent two-group comparisons) Definition: A non-parametric test (distribution-free) used to compare more than two independent groups of sampled data. Test: The hypotheses for the comparison of independent groups are: H 0 : The samples of all groups come from identical populations H a : The samples of all groups come from different populations HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 34 34

35 Kruskall-Wallis test i... k X 11 X X 1i... X 1k X 21 X X 2i... X 2k Xn 2 2 Xn i i Xn k k Xn 1 1 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 35 35

36 Kruskall-Wallis test x jj =μ+τ i +ε ij, j=1,2,..., n i, i=1,2,..., k and N=Σn i. (i=1,2,...k) where μ is the unknown expected value τ i is the effect of ith treatment. H 0 : τ 1 = τ 2 =...= τ k H A : τ o τ p, there is at least one group differs from others. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 36 36

37 Kruskall-Wallis test Combine and sort all x ij values in ascending order. r ij denotes the rank of x ij. We know: n i R i = r ji i R. i = j= 1 n k i R.. = k i = 1 R i = i i= 1 N N N( N + R 2 = 1) HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach R

38 Test Statistics H = 12 N( N + 1) k i = 1 n i 2 12 R = i ( R. i R.. ) 3( N + 1) N( N + 1) n k i = 1 2 i H statistics is approximately chi-square distributed with k-1 degrees of freedom HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 38 38

39 Example We have results of three treatments A B C HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 39 39

40 Assign ranks A B C A B C average of ranks H = 12 21(21 + 1) 131 ( ) 3(21 + 1) =

41 STATA Result for Kruskal-Wallis test Test: Equality of populations (Kruskal-Wallis Test) Groups _Obs _RankSum chi-squared = with 2 d.f. probability = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 41 41

42 Spearman's rank correlation coefficient The rank correlation coefficient is the Pearson correlation coefficient based on the ranks of the data if there are no ties (adjustments are made if some of the data are tied). If the original data for each variable have no ties. the data for each variable are first ranked. and then the Pearson correlation coefficient between the ranks for the two variables is computed. Like Pearson correlation coefficient. the rank correlation ranges between -1 and +1. where -1 and +1 indicate a perfect linear relationship between the ranks of the two variables. The interpretation is therefore the same except that the relationship between ranks. and not values. is examined HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 42 42

43 Ranks of the 1.sample Ranks of the 2.sample Difference r 1 q 1 d 1 =r 1 -q 1 r 2 q 2 d 2 =r 2 -q r n q n d n =r n -q n HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 43 43

44 Test statistics r s = 1 6 n n i = 1 3 d 2 i n r n 2 t = 1 r 2 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 44 44

45 The t-test H 0 : correlation coefficient in population = 0, in notation: ρ =0 H a : ρ 0 This test can be carried out by expressing the t statistic in terms of r. It can be proven that the statistic has t-distribution with n-2 degrees of freedom t = r 1 n 2 r = r n 2 1 r 2 2 Decision using statistical table: If t table denotes the value of the table corresponding to n-2 degrees of freedom and probability, if t > t table, we reject H 0 and state that the population correlation coefficient,ρ is different from 0. Decision using p-value: if p < α (=0.05) we reject H 0 and state that the population correlation coefficient, ρ is different from 0 45 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 45

46 Example for Spaerman rank correlation The effectiveness of a treatment was measured on a scale between The scores were determined by both the patients and doctors. Is there any relationship between the patients and doctors scores? HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 46 46

47 Data patient doctor HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 47 47

48 The results patients doctors Rank (patients ) Rank (doctor) difference d i HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 48 48

49 Results H 0 : correlation coefficient in population = 0, in notation: ρ =0 H a : ρ 0 r s = 1 6* n n i = 1 3 d n i 6*62 = 1 = t = r n r = = 2.26 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 49 49

50 Number of obs = 10 Spearman's rho = STATA results Test of Ho: patient and doctor independent Pr > t = HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 50 50

51 Jonckheere-Terpstra Test (JP) The Jonckheere-Terpstra test. which is a nonparametric test for ordered differences among classes. It tests the null hypothesis that the distribution of the response variable does not differ among classes. It is designed to detect alternatives of ordered class differences. which can be expressed as (or ). with at least one of the inequalities being strict. where denotes the effect of class i. For such ordered alternatives. the Jonckheere-Terpstra test can be preferable to tests of more general class difference alternatives. such as the Kruskal - Wallis test. The Jonckheere-Terpstra test is appropriate for a contingency table in which an ordinal column variable represents the response. The row variable. which can be nominal or ordinal. represents the classification variable. The levels of the row variable should be ordered according to the ordering you want the test to detect HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 51 51

52 Jonckheere-Terpstra statistics The Jonckheere-Terpstra test statistic is computed by first forming R(R-1)/2 Mann- Whitney counts M i.i'. where i < i'. for pairs of rows in the contingency table. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 52 52

53 Null and alternative hypothesis H 0 : 1 2 τ = τ =... = τ k H A : τ 1 τ 2... τ k HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 53 53

54 Test statistics T uv δ = = n u n v i = 1 i = 1 ' δ ( X iu, X 1, if a < b 1 {, if a = b 2 0, otherwise i' v ) J = u < v k 1 k T uv = T u = 1 v= 1 uv HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 54 54

55 Example:Do five different chemotherapy methods differ significantly in treatment response? A small pilot study was performed with five chemotherapy regimens: Cytoxan (CTX) alone, Cyclohexyl-chloreoethyl nitrosourea (CCNU) alone, Methotrexate (MTX) alone, CTX and MTX together, and CTX, CCNU, and MTX together. Tumor regression was measured on a threepoint scale: no response, partial response, and complete response. The results are displayed in the following Table. HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 55 55

56 Example No. of Patients Chemo No Response Partial Response Complete Response CTX CCNU MTX CTX+CCNU CTX+CCNU+MTX HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 56 56

57 Chemo Ranks No Response No. of Patients Partial Response Complete Response CTX 12 3,5 3,5 CCNU 8,5 8,5 3,5 MTX 14 3,5 3,5 CTX+CCNU ,5 CTX+CCNU+MTX 8,5 8,5 15 HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 57 57

58 T 12 n δ (2,0) + n 1 2 = i = 1 i = 1 ' Test statistics δ ( X i 1, X i = δ (2,2) + δ (1,0) + δ (1,1) + δ (1,0) + δ (1,2) + δ + δ (1,0) + δ (1,1) + δ (1,0) + δ (1,2) + '2 ) δ (2,1) + δ (2,0) + T 20 T = = δ (2,1) + (1,1) +... δ (1,1) = 20 J = 56 = 54 J critical HUSRB/0901/221/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided Approach 58 58